The Tracy-Widom Distribution at Large Dyson Index

We study the Tracy-Widom (TW) distribution f_β (a) f β ( a ) in the limit of large Dyson index β → +∞ β → + ∞ . This distribution describes the fluctuations of the rescaled largest eigenvalue a₁ a 1 of the Gaussian (alias Hermite) ensemble (G β β E) of (infinitely) large random matrices. We show that, at large β β , its probability density function takes the large deviation form f_β (a) ~ e-β Φ (a) f β ( a ) ∼ e - β Φ ( a ) . While the typical deviation of a₁ a 1 around its mean is Gaussian of variance O(1/β ) O ( 1 / β ) , this large deviation form describes the probability of rare events with deviation O (1), and governs the behavior of the higher cumulants. We obtain the rate function Φ (a) Φ ( a ) as a solution of a Painlevé II equation. We derive explicit formula for its large argument behavior, and for the lowest cumulants, up to order 4. We compute Φ (a) Φ ( a ) numerically for all a and compare with exact numerical computations of the TW distribution at finite β β . These results are obtained by applying saddle-point approximations to an associated problem of energy levels $$E=-a$$ E = - a , for a random quantum Hamiltonian defined by the stochastic Airy operator (SAO). We employ two complementary approaches: (i) we use the optimal fluctuation method to find the most likely realization of the noise in the SAO, conditioned on its ground-state energy being E (ii) we apply the weak-noise theory to the representation of the TW distribution in terms of a Ricatti diffusion process associated to the SAO. We extend our results to the full Airy point process a₁>a₂> <mml:m